Monotonicity and the Dominated Farthest Points Problem in Banach Lattice

Abstract

We introduce the dominated farthest points problem in Banach lattices. We prove that for two equivalent norms such that X becomes an STM and LLUM space the dominated farthest points problem has the same solution. We give some conditions such that under these conditions the Fréchet differentiability of the farthest point map is equivalent to the continuity of metric antiprojection in the dominated farthest points problem. Also we prove that these conditions are equivalent to strong solvability of the dominated farthest points problem. We prove these results in STM, reflexive STM, and UM spaces. Moreover, we give some applications of the stated results in Musielak-Orlicz spaces $L^{\varphi }(\mu )$ and $E^{\varphi }(\mu )$ over nonatomic measure spaces in terms of the function $\varphi$. We will prove that the Fréchet differentiability of the farthest point map and the conditions $\varphi \in {\mathrm{\Delta }}_2$ and $\varphi >0$ in reflexive Musielak-Orlicz function spaces are equivalent.